| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2017 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Product with unknown constant to determine |
| Difficulty | Standard +0.3 Part (i) is a standard binomial expansion requiring identification of the constant term (r=2 gives x^0). Part (ii) adds a simple algebraic step: multiplying by (1+ax²) shifts powers, requiring students to set up an equation where terms cancel to eliminate x^0. This is slightly above routine but still a textbook-style exercise with clear methodology. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(6C3\left(\frac{2}{x}\right)^3(-3x)^3\) SOI, also allowed if seen in an expansion | M1 | Both \(x\)'s can be missing |
| \(-4320\) Identified as answer | A1 | Cannot be earned retrospectively in (ii) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(6C2\left(\frac{2}{x}\right)^4[(-3x)]^2\) SOI clearly identified as critical term | M1 | Both \(x\)'s and minus sign can be missing |
| \(15a \times 16 \times 9 - their\ 4320\ (=0)\) | A1 FT | FT on *their* 4320 |
| \(a=2\) | A1 |
## Question 3(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $6C3\left(\frac{2}{x}\right)^3(-3x)^3$ SOI, also allowed if seen in an expansion | M1 | Both $x$'s can be missing |
| $-4320$ Identified as answer | A1 | Cannot be earned retrospectively in (ii) |
## Question 3(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $6C2\left(\frac{2}{x}\right)^4[(-3x)]^2$ SOI clearly identified as critical term | M1 | Both $x$'s and minus sign can be missing |
| $15a \times 16 \times 9 - their\ 4320\ (=0)$ | A1 FT | FT on *their* 4320 |
| $a=2$ | A1 | |
---3 (i) Find the term independent of $x$ in the expansion of $\left( \frac { 2 } { x } - 3 x \right) ^ { 6 }$.\\
(ii) Find the value of $a$ for which there is no term independent of $x$ in the expansion of
$$\left( 1 + a x ^ { 2 } \right) \left( \frac { 2 } { x } - 3 x \right) ^ { 6 }$$
\hfill \mbox{\textit{CAIE P1 2017 Q3 [5]}}